Zeus was very powerful and quite macho, so he had an extremely active love life (wait, no, even that's an understatement). Everyone wanted his attention, and an unsurprising number of lucky goddesses and women got it. Some of these relationships were rather interesting, especially since Zeus's ever-jealous wife, Hera, regularly meddled with his lovers and offspring...
Metis was a second-generation Titaness (the Titans were the ruling gods before the Olympians), the Titaness of cunning. She was also Zeus's first legitimate wife, before Hera. A prophecy had warned Zeus, that, like his father and grandfather before him, his wife would bear a child who would overthrow him. So Zeus changed Metis into a fly and swallowed her. As such, Metis wouldn't bear the child - he would.
Meanwhile, inside Zeus's brain, Metis took on a miniature form of herself, and started using her hammer on some trusty scrap metal she happened to keep with her. She banged away, creating armor for her little unborn goddess. This gave Zeus the Worst. Migraine. Ever. Remember, aspirin hadn't been discovered yet, so Zeus was naturally inclined to crack his head open. A few months later, he did (and being a god, he survived). Out popped mini-Metis, as well as Athena, goddess of wisdom, strategy, and housekeeping.
Leto was another second-generation Titaness; Hera liked Zeus's relationship with this poor goddess the least of any of them. After learning Leto was pregnant, Hera forbade her from giving birth on unmoving land. (There was a catch, though: there were some floating islands in the Aegean Sea, so that went off without a hitch.) Next, Hera had her daughter Eileithyia, the goddess of childbirth, cross her legs and thus keep Leto in labor. (Zeus had Hypnos, god of sleep, knock Eileithyia out and uncross her legs. Crisis averted.) Finally, Leto gave birth to Olympians Artemis (goddess of hunting and the Moon) and Apollo (god of the Sun, literature, and music) on the floating island of Delos. But Hera didn't stop there. She sent a huge snake, Python, to scare Leto. Apollo, though still a baby, killed Python. Hera finally gave up and let Artemis and Apollo into Olympus.
Leda was the daughter of the king of Aetolia; Zeus visited her as a swan. Since Leda had just "done it" with her husband, she gave birth to two sets of twins, one fathered by each man, and each set contained in a large egg. One egg contained Pollux and Helen (the beauty of the Trojan War), who were immortal children of Zeus, and the other contained Castor and Clytemnestra, who were the mortal children of Leda's husband.
Europa was a princess of Phoenicia; Zeus approached her as a white bull. Europa was astonished at how calm the bull was, and jumped on his back. He promptly carried her off to Crete, where, nine months later, she gave birth to triplets: Minos, Aeacus, and Rhadamanthus, all famous kings who became judges of the Underworld after their deaths. (Minos, however, didn't really play fair...)
Semele, another princess of Phoenicia, was tricked by Hera (in the disguise of her nurse) into asking Zeus to show himself to her in his true form. The problem with that? His true form was a deadly flash of epic godly lightning. Unwillingly, Zeus obeyed Semele's orders, charring the girl almost beyond recognition. Then, Zeus remembered that Semele was six months pregnant, took his child out of her, and sewed the embryo into his thigh. After 3 more months, Zeus took the baby out and brought him to the nymphs at Nysa, who named him Dionysus.
Callisto was a princess of Arcadia, and a huntress at the side of Artemis. She had sworn to remain a virgin, so when Zeus showed up and "did it" with her, Artemis was furious and turned Callisto into a bear, but not before she'd had a son, Arcas. Fifteen years later, Arcas spotted bear-Callisto while hunting and almost killed her, but Zeus changed Arcas into a bear cub. After their deaths, bear-Callisto and bear-Arcas became - you guessed it! - the Big Bear and Little Bear constellations.
Alcmene, the princess of Mycenae, was "visited" by Zeus while he was disguised as her husband. She ended up having twins from that partnership: Heracles/Hercules, son of Zeus, and Iphicles, son of her real husband. Hera hated Heracles (whose name actually means "famous because of Hera"), so when he was a baby, Hera sent two snakes to kill the tot. Of course, Alcmene must have freaked out when she found him laughing with the dead snakes that he had strangled in his hands! Hera continued to plague her stepson throughout his life.
Io, a princess of Argos and priestess of Hera, was temporarily transformed by Zeus into a white cow in order to try to trick Hera into ignoring her, as the white cow was one of her sacred animals. Hera wasn't fooled. She put cow-Io under the protection of Argus-Panoptes, a monster with 100 eyes. At Zeus's request, Hermes played music on his new lyre and put the beast to sleep before killing it. Cow-Io was temporarily free, but Hera didn't stop there. She sent a horsefly to annoy Io, who then wandered to Egypt. On the banks of the Nile, Io changed back into a human and gave birth to Epaphus, who became the king of Egypt.
Danae was another princess of Argos. Her father, Acrisius, heard a prophecy that his grandson would kill him, so he imprisoned Danae in a tower. Zeus visited her concealed in a drop of golden rain, and they "did it" and had a baby, Perseus.